1. Stating the problem: We have the cubic function $$F(x) = -x^3 + 2x^2 - 9x - 15$$. 2. From the table and given values, it seems we want to analyze the roots or evaluate the function at specific points. 3. To find roots, let's check for possible rational roots using the Rational Root Theorem: possible roots are factors of 15, i.e., $\pm1, \pm3, \pm5, \pm15$. 4. Evaluate $F(5)$: $$F(5) = -(5)^3 + 2(5)^2 - 9(5) - 15 = -125 + 50 - 45 - 15 = -135.$$ So, 5 is not a root. 5. Evaluate $F(-3)$: $$F(-3) = -(-3)^3 + 2(-3)^2 - 9(-3) - 15 = -(-27) + 2(9) + 27 - 15 = 27 + 18 + 27 - 15 = 57.$$ Not zero, so no root. 6. Evaluate $F(3)$: $$F(3) = -(3)^3 + 2(3)^2 - 9(3) - 15 = -27 + 18 - 27 -15 = -51.$$ Not zero. 7. Evaluate $F(1)$: $$F(1) = -1 + 2 - 9 - 15 = -23.$$ Not zero. 8. Evaluate $F(-1)$: $$F(-1) = -(-1) + 2(1) + 9 - 15 = 1 + 2 + 9 - 15 = -3.$$ Not zero. 9. Since none of these simple values gave zero, we consider the given table values may show intermediate calculations or factors. 10. Let's try to factor $F(x)$ using polynomial division with possible roots: Try dividing by $(x+3)$: Synthetic division coefficients: -1, 2, -9, -15 Using -3: Carry down -1, multiply by -3 = 3, add to 2 = 5 Multiply 5 by -3 = -15, add to -9 = -24 Multiply -24 by -3 = 72, add to -15 = 57 Remainder is 57, so not a root. Try dividing by $(x-5)$: Using 5: Carry down -1, multiply -1*5 = -5, add to 2 = -3 Multiply -3*5 = -15, add to -9 = -24 Multiply -24*5 = -120, add to -15 = -135 Remainder -135, no root. 11. The expression cannot be factored easily with integer roots, so the function has no rational roots. 12. Final conclusion: The cubic $F(x)$ has no simple rational roots among candidates tested. Final answer: The function $F(x) = -x^3 + 2x^2 - 9x - 15$ does not have rational roots from the candidates tested.